Normal Distribution

Normal Distribution
It is important because of Central Limit Theorem (CTL), the CTL said that Sum up a lot of i.i.d random variables the shape of the distribution will looks like Normal.
Normal P.D.F Now we want to find c This integral has been proved that it cannot have close form solution. However, someone gives an idea that looks stupid but actually very brilliant by multiply two of them.

reminds the function of circle which we can replace them to polar coordinate

Thus
Mean

By symmetry if g(x) is odd function g-x=-g(x) then -abgxdx=0
Variance

Notation
CDF is standard Normal CDF by symmetric

,CDF , , All the odd moment of standard normal are zero. However, even moment is not easy to calculate by integral (Symmetry) Then we say
Most of Statistics books will write the pdf then explain the mean and variance but it is not intuitive. Standardization Find PDF of
CDF:
The PDF is derivative of the CDF (using chain rule)
PDF:

Later we’ll show if independent 68-95-99.7% Rule
Because you can’t actually calculate the , somebody create a rule of thumb The properties of variance If you shift the variance by c, the mean also shift by c. Thus, the variance doesn’t change. Remember to square. It is easy to validate if you think c is negative and don’t square it. The corresponding variance becomes negative which contradict the definition of the variance. iff for some
Variance only can be zero if and only if the probability is a constant. in general [It is equal if X,Y are independent]
To get the idea about this we let Y = X which means the two r.v.s are extremely dependent. Then